Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-hbs1 Structured version   Visualization version   GIF version

Theorem bj-hbs1 31946
 Description: Version of hbsb2 2347 with a dv condition, which does not require ax-13 2234, and removal of ax-13 2234 from hbs1 2424. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbs1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-hbs1
StepHypRef Expression
1 bj-sb4v 31945 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 bj-sb2v 31941 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
32axc4i 2116 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
41, 3syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  bj-nfs1v  31947  bj-hbab1  31959
 Copyright terms: Public domain W3C validator